Further popular approaches for solving constrained problems are the Augmented Lagrangian method and ADMM (Alternating Direction Method of Multipliers)
[1]}, [2]}, [3]}, [4]}. Such methods work directly on the Lagrangian formulation of the problem while adding penalty terms.
Under specific conditions, their convergence rate may be linear [5]}, [6]}. However,
ADMM requires the ability to efficiently compute the proximal operator, which as a special case includes the projection operator.
To project onto the intersection of convex constraint sets \(\cap _{i=1}^m \mathcal {K}_i\) , consensus ADMM can exploit projection oracles for each \(\mathcal {K}_i\)  separately. For this general case, only sublinear rate is shown [7]}, [8]}. In the special case of polyhedral sets \(\mathcal {K}_i\) , it can have linear rate [9]}.
